3.478 \(\int \frac{-7+4 x^2}{3+2 x} \, dx\)

Optimal. Leaf size=13 \[ x^2-3 x+\log (2 x+3) \]

[Out]

-3*x + x^2 + Log[3 + 2*x]

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Rubi [A]  time = 0.010276, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {697} \[ x^2-3 x+\log (2 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(-7 + 4*x^2)/(3 + 2*x),x]

[Out]

-3*x + x^2 + Log[3 + 2*x]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{-7+4 x^2}{3+2 x} \, dx &=\int \left (-3+2 x+\frac{2}{3+2 x}\right ) \, dx\\ &=-3 x+x^2+\log (3+2 x)\\ \end{align*}

Mathematica [A]  time = 0.0033279, size = 16, normalized size = 1.23 \[ x^2-3 x+\log (2 x+3)-\frac{27}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(-7 + 4*x^2)/(3 + 2*x),x]

[Out]

-27/4 - 3*x + x^2 + Log[3 + 2*x]

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Maple [A]  time = 0.002, size = 14, normalized size = 1.1 \begin{align*} -3\,x+{x}^{2}+\ln \left ( 3+2\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2-7)/(3+2*x),x)

[Out]

-3*x+x^2+ln(3+2*x)

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Maxima [A]  time = 1.15183, size = 18, normalized size = 1.38 \begin{align*} x^{2} - 3 \, x + \log \left (2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="maxima")

[Out]

x^2 - 3*x + log(2*x + 3)

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Fricas [A]  time = 1.22185, size = 35, normalized size = 2.69 \begin{align*} x^{2} - 3 \, x + \log \left (2 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="fricas")

[Out]

x^2 - 3*x + log(2*x + 3)

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Sympy [A]  time = 0.071488, size = 12, normalized size = 0.92 \begin{align*} x^{2} - 3 x + \log{\left (2 x + 3 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2-7)/(3+2*x),x)

[Out]

x**2 - 3*x + log(2*x + 3)

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Giac [A]  time = 1.12069, size = 19, normalized size = 1.46 \begin{align*} x^{2} - 3 \, x + \log \left ({\left | 2 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="giac")

[Out]

x^2 - 3*x + log(abs(2*x + 3))